For a fixed partition $\lambda=(\lambda_1\geq\dots\geq \lambda_n)$ we say $\mu=(\mu_1\geq \dots \geq \mu_{n-1})$ $\textit{interlaces}$ $\lambda$ iff $$\lambda_1\geq \mu_1\geq \dots \geq \mu_{n-1}\geq \lambda_n .$$ Is there a way of counting how many partitions interlace a given partition $\lambda$?
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asked Nov 8, 2022 at 15:38
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$\begingroup$ If we use $\mu \prec \lambda$ to denote "$\mu$ interlaces $\lambda$," then it is well-known that the number of semistandard Young tableaux of shape lambda with entries in $\{1,2,\ldots,n\}$ is the number of sequences $\lambda^{0} \prec \lambda^{1} \prec \cdots \prec \lambda^{n} = \lambda$. Each $\lambda^{i-1} \prec \lambda^{i}$ determines a horizontal strip of where the $i$'s in the SSYT are. You're basically asking for the "first step" of this process: the number of ways to place $n$ in an SSYT of shape $\lambda$. There should be a determinantal formula for this, at a minimum. $\endgroup$– Sam HopkinsCommented Nov 8, 2022 at 15:53
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2$\begingroup$ Isn't this $\prod_{j=1}^{n-1} (\lambda_j - \lambda_{j+1}+1)$? Because $\lambda_j - \lambda_{j+1}+1$ is the number of choices for $\mu_j$, and they can all be chosen independently. (PS If you want to exactly match SSYT, you should also allow $\mu$ to have an $n$-th part, with $\lambda_n \geq \mu_n \geq 0$.) $\endgroup$– David E SpeyerCommented Nov 8, 2022 at 16:08
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1$\begingroup$ @DavidESpeyer: Your observation about the product formula is of course correct. But regarding SSYT, I do think we want $\mu$ to have $n-1$ entries, as in the question-asker's post. For example, think of the way Gelfand-Tsetlin patterns are triangular: each row has one less entry than the previous one. $\endgroup$– Sam HopkinsCommented Nov 8, 2022 at 16:14
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$\begingroup$ Oh, you are right. I can think of contexts in which I would be right, but the most straightforward thing is what @SamHopkins said. $\endgroup$– David E SpeyerCommented Nov 8, 2022 at 17:49
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$\begingroup$ Link to Gelfand-Tsetlin pattern definition: symmetricfunctions.com/gtpatterns.htm (and how they relate to SSYTs) $\endgroup$– Per AlexanderssonCommented Nov 14, 2022 at 6:47
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